The purpose of this paper is to characterize the homogeneous Besov space in the Dunkl setting. We utilize a new discrete reproducing formula, that is, the building blocks are differences of the Dunkl-Poisson kernel which involves both the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the Dunkl setting, new test functions and distributions are introduced, and a new decomposition is established.
{"title":"Homogeneous Besov Space in Dunkl setting","authors":"Mengmeng Dou, Jiashu Zhang","doi":"arxiv-2408.00340","DOIUrl":"https://doi.org/arxiv-2408.00340","url":null,"abstract":"The purpose of this paper is to characterize the homogeneous Besov space in\u0000the Dunkl setting. We utilize a new discrete reproducing formula, that is, the\u0000building blocks are differences of the Dunkl-Poisson kernel which involves both\u0000the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the\u0000Dunkl setting, new test functions and distributions are introduced, and a new\u0000decomposition is established.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is shown that geometric Brascamp--Lieb data are, in a certain sense, ubiquitous. This addresses a question raised by Bennett and Tao in their recent work on the adjoint Brascamp--Lieb inequality.
本论文主要依据 Garg、Gurvits、Oliveira 和 Wigderson 的研究成果，证明了几何布拉什坎普--勒布数据在某种意义上是无处不在的，从而解决了 Bennett 和 Tao 在他们最近关于联合布拉什坎普--勒布不等式的研究中提出的一个问题。
{"title":"A note on ubiquity of geometric Brascamp-Lieb data","authors":"Neal Bez, Anthony Gauvan, Hiroshi Tsuji","doi":"arxiv-2407.21440","DOIUrl":"https://doi.org/arxiv-2407.21440","url":null,"abstract":"Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is\u0000shown that geometric Brascamp--Lieb data are, in a certain sense, ubiquitous.\u0000This addresses a question raised by Bennett and Tao in their recent work on the\u0000adjoint Brascamp--Lieb inequality.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see when and how any given sequence of polynomials is Darboux related to a diagonal matrix of classic orthogonal polynomials. We also explore the notion of Darboux-irreducibility and study some sequences that are not a Darboux transformation of classical orthogonal polynomials.
{"title":"Darboux equivalence for matrix-valued orthogonal polynomials","authors":"Ignacio Bono Parisi, Inés Pacharoni, Ignacio Zurrián","doi":"arxiv-2407.20994","DOIUrl":"https://doi.org/arxiv-2407.20994","url":null,"abstract":"In this work, we give some criteria that allow us to decide when two\u0000sequences of matrix-valued orthogonal polynomials are related via a Darboux\u0000transformation and to build explicitly such transformation. In particular, they\u0000allow us to see when and how any given sequence of polynomials is Darboux\u0000related to a diagonal matrix of classic orthogonal polynomials. We also explore\u0000the notion of Darboux-irreducibility and study some sequences that are not a\u0000Darboux transformation of classical orthogonal polynomials.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We demonstrate that the almost everywhere convergence of the planar Bochner-Riesz means for $L^p$ functions in the optimal range when $5/3leq pleq 2$. This is achieved by establishing a sharp $L^{5/3}$ estimate for a maximal operator closely associated with the Bochner-Riesz multiplier operator. The estimate depends on a novel refined $L^2$ estimate, which may be of independent interest.
我们证明，当 $5/3leqpleq 2$ 时，L^p$ 函数的 PlanarBochner-Riesz means 几乎无处不收敛。这是通过为与波赫纳-里兹乘法算子密切相关的最大算子建立一个尖锐的$L^{5/3}$估计值来实现的。
{"title":"On almost everywhere convergence of planar Bochner-Riesz mean","authors":"Xiaochun Li, Shukun Wu","doi":"arxiv-2407.20887","DOIUrl":"https://doi.org/arxiv-2407.20887","url":null,"abstract":"We demonstrate that the almost everywhere convergence of the planar\u0000Bochner-Riesz means for $L^p$ functions in the optimal range when $5/3leq\u0000pleq 2$. This is achieved by establishing a sharp $L^{5/3}$ estimate for a\u0000maximal operator closely associated with the Bochner-Riesz multiplier operator.\u0000The estimate depends on a novel refined $L^2$ estimate, which may be of\u0000independent interest.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish Triebel-Lizorkin spaces in the Dunkl setting which are associated with finite reflection groups on the Euclidean space. The group structures induce two nonequivalent metrics: the Euclidean metric and the Dunkl metric. In this paper, the L^2 space and the Dunkl-Calderon-Zygmund singular integral operator in the Dunkl setting play a fundamental role. The main tools used in this paper are as follows: (i) the Dunkl-Calder'on-Zygmund singular integral operator and a new Calderon reproducing formula in L^2 with the Triebel-Lizorkin space norms; (ii) new test functions in terms of the L^2 functions and distributions; (iii) the Triebel-Lizorkin spaces in the Dunkl setting which are defined by the wavelet-type decomposition with norms and the analogous atomic decomposition of the Hardy spaces.
{"title":"Triebel-Lizorkin spaces in Dunkl setting","authors":"Chuhan Sun, Zhiming Wang","doi":"arxiv-2408.05227","DOIUrl":"https://doi.org/arxiv-2408.05227","url":null,"abstract":"We establish Triebel-Lizorkin spaces in the Dunkl setting which are\u0000associated with finite reflection groups on the Euclidean space. The group\u0000structures induce two nonequivalent metrics: the Euclidean metric and the Dunkl\u0000metric. In this paper, the L^2 space and the Dunkl-Calderon-Zygmund singular\u0000integral operator in the Dunkl setting play a fundamental role. The main tools\u0000used in this paper are as follows: (i) the Dunkl-Calder'on-Zygmund singular\u0000integral operator and a new Calderon reproducing formula in L^2 with the\u0000Triebel-Lizorkin space norms; (ii) new test functions in terms of the L^2\u0000functions and distributions; (iii) the Triebel-Lizorkin spaces in the Dunkl\u0000setting which are defined by the wavelet-type decomposition with norms and the\u0000analogous atomic decomposition of the Hardy spaces.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the dimension of planar elliptic measures via the application of quasiconformal mappings. In fact, in our case studies, we find a quasiconformal mapping that relates the elliptic measure in a domain to the harmonic measure in its image domain, and we deduce bounds for the Hausdorff dimension of the elliptic measure by the known results on the harmonic side.
{"title":"The Hausdorff dimension of planar elliptic measures via quasiconformal mappings","authors":"Ignasi Guillén-Mola","doi":"arxiv-2407.21145","DOIUrl":"https://doi.org/arxiv-2407.21145","url":null,"abstract":"In this paper we study the dimension of planar elliptic measures via the\u0000application of quasiconformal mappings. In fact, in our case studies, we find a\u0000quasiconformal mapping that relates the elliptic measure in a domain to the\u0000harmonic measure in its image domain, and we deduce bounds for the Hausdorff\u0000dimension of the elliptic measure by the known results on the harmonic side.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we gather and extend classical results for parabolic cylinder functions, namely solutions of the Weber differential equations, using a systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of the Weber differential equation begin{equation*} w''(z)-left(frac{z^2}{4}+aright)w(z)=0 end{equation*} and provide representations by Laplace integrals extended to include all values of the complex parameter $a$; we find an integral integral representation for the function $V$; none was previously available. For the Weber equation in the form begin{equation*} u''(x)+left(frac{x^2}{4}-aright)u(x)=0, end{equation*} we define a new fundamental system $E_pm$ which is analytic in $ainmathbb{C}$, based on asymptotic behavior; they appropriately extend and modify the classical solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for the approach.
{"title":"Parabolic cylinder functions revisited using the Laplace transform","authors":"Rodica D. Costin, Georgios Mavrogiannis","doi":"arxiv-2407.20403","DOIUrl":"https://doi.org/arxiv-2407.20403","url":null,"abstract":"In this paper we gather and extend classical results for parabolic cylinder\u0000functions, namely solutions of the Weber differential equations, using a\u0000systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of\u0000the Weber differential equation begin{equation*}\u0000w''(z)-left(frac{z^2}{4}+aright)w(z)=0 end{equation*} and provide\u0000representations by Laplace integrals extended to include all values of the\u0000complex parameter $a$; we find an integral integral representation for the\u0000function $V$; none was previously available. For the Weber equation in the form begin{equation*} u''(x)+left(frac{x^2}{4}-aright)u(x)=0, end{equation*} we define a new\u0000fundamental system $E_pm$ which is analytic in $ainmathbb{C}$, based on\u0000asymptotic behavior; they appropriately extend and modify the classical\u0000solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for\u0000the approach.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the testing condition for Radon-Brascamp-Lieb multilinear functionals established in arXiv:2201.12201, this paper is concerned with identifying local conditions on smooth maps $u(t)$ with values in the space of decomposable p-forms on some real vector space V which guarantee uniform integrability of $||u(t)||^{-tau}$ over a certain natural, noncompact family of norms. One can loosely regard this problem as a higher-dimensional analogue of establishing uniform bounds for the size of a sublevel set of a function in terms of the size of its derivatives. The resulting theorem relies extensively on ideas from Geometric Invariant Theory to understand what appropriate derivative bounds look like in this context. Several examples and applications are presented, including a new local characterization of so-called "model" Radon-like transforms in terms of the semistability of a natural curvature functional (giving an equivalent but rather different criterion than the one first established in arXiv:2303.03325).
{"title":"Generalized Sublevel Estimates for Form-Valued Functions and Related Results for Radon-like Transforms","authors":"Philip T. Gressman","doi":"arxiv-2407.18860","DOIUrl":"https://doi.org/arxiv-2407.18860","url":null,"abstract":"Motivated by the testing condition for Radon-Brascamp-Lieb multilinear\u0000functionals established in arXiv:2201.12201, this paper is concerned with\u0000identifying local conditions on smooth maps $u(t)$ with values in the space of\u0000decomposable p-forms on some real vector space V which guarantee uniform\u0000integrability of $||u(t)||^{-tau}$ over a certain natural, noncompact family\u0000of norms. One can loosely regard this problem as a higher-dimensional analogue\u0000of establishing uniform bounds for the size of a sublevel set of a function in\u0000terms of the size of its derivatives. The resulting theorem relies extensively\u0000on ideas from Geometric Invariant Theory to understand what appropriate\u0000derivative bounds look like in this context. Several examples and applications\u0000are presented, including a new local characterization of so-called \"model\"\u0000Radon-like transforms in terms of the semistability of a natural curvature\u0000functional (giving an equivalent but rather different criterion than the one\u0000first established in arXiv:2303.03325).","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give a quantitative stability result for the discrete interaction energy on the multi-dimensional torus, for the periodic Riesz potential. It states that if the number of particles $N$ is large and the discrete interaction energy is low, then the particle distribution is necessarily close to the uniform distribution (i.e., the continuous energy minimizer) in the Wasserstein-infinity distance. As a consequence, we obtain a quantitative mean field limit of interaction energy minimizers in the Wasserstein-infinity distance. The proof is based on the application of the author's previous joint work with J. Wang on the stability of continuous energy minimizer, together with a new mollification trick for the empirical measure in the case of singular interaction potentials.
{"title":"Wasserstein-infinity stability and mean field limit of discrete interaction energy minimizers","authors":"Ruiwen Shu","doi":"arxiv-2407.18395","DOIUrl":"https://doi.org/arxiv-2407.18395","url":null,"abstract":"In this paper we give a quantitative stability result for the discrete\u0000interaction energy on the multi-dimensional torus, for the periodic Riesz\u0000potential. It states that if the number of particles $N$ is large and the\u0000discrete interaction energy is low, then the particle distribution is\u0000necessarily close to the uniform distribution (i.e., the continuous energy\u0000minimizer) in the Wasserstein-infinity distance. As a consequence, we obtain a\u0000quantitative mean field limit of interaction energy minimizers in the\u0000Wasserstein-infinity distance. The proof is based on the application of the\u0000author's previous joint work with J. Wang on the stability of continuous energy\u0000minimizer, together with a new mollification trick for the empirical measure in\u0000the case of singular interaction potentials.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce the fractional medians, give an expression of the set of all fractional medians in terms of non-increasing rearrangements and then investigate mapping properties of the fractional maximal operators defined by such medians. The maximal operator is a generalization of that in Stromberg. It turns out that our maximal operator is a more smooth operator than the usual fractional maximal operator. Further, we give another proof of the embedding from $BV$ to $L^{n/(n-1),1}$ due to Alvino by using the usual medians.
{"title":"Fractional medians and their maximal functions","authors":"Yohei Tsutsui","doi":"arxiv-2407.17700","DOIUrl":"https://doi.org/arxiv-2407.17700","url":null,"abstract":"In this article, we introduce the fractional medians, give an expression of\u0000the set of all fractional medians in terms of non-increasing rearrangements and\u0000then investigate mapping properties of the fractional maximal operators defined\u0000by such medians. The maximal operator is a generalization of that in Stromberg.\u0000It turns out that our maximal operator is a more smooth operator than the usual\u0000fractional maximal operator. Further, we give another proof of the embedding\u0000from $BV$ to $L^{n/(n-1),1}$ due to Alvino by using the usual medians.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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